Some consequences of shear on galactic dynamos with helicity fluxes

TL;DR

This paper extends galactic dynamo models by incorporating shear effects on turbulence energy and correlation time, revealing how shear influences magnetic field saturation and helicity fluxes in galaxy evolution.

Contribution

It introduces an analytic framework accounting for shear impacts on turbulence and helicity fluxes, including magnetic buoyancy, with comparisons to rotation-only models.

Findings

Shear significantly reduces turbulence correlation time at low Rossby numbers.

Magnetic buoyancy can sustain helicity fluxes without winds or diffusion.

Shear effects alter dynamo saturation levels depending on turbulence timescales.

Abstract

Galactic dynamo models sustained by supernova (SN) driven turbulence and differential rotation have revealed that the sustenance of large scale fields requires a flux of small scale magnetic helicity to be viable. Here we generalize a minimalist analytic version of such galactic dynamos to explore some heretofore unincluded contributions from shear on the total turbulent energy and turbulent correlation time, with the helicity fluxes maintained by either winds, diffusion, or magnetic buoyancy. We construct an analytic framework for modeling the turbulent energy and correlation time as functions of SN rate and shear. We compare our prescription with previous approaches that only include rotation. The solutions depend separately on the rotation period and the eddy turnover time and not just on their ratio (the Rossby number). We consider models in which these two time scales are allowed to be independent and also a case in which they are mutually dependent on radius when a radial dependent SN rate model is invoked. For the case of a fixed rotation period (or fixed radius) we show that the influence of shear is dramatic for low Rossby numbers, reducing the correlation time of the turbulence, which in turn, strongly reduces the saturation value of the dynamo compared to the case when the shear is ignored. We also show that even in the absence of winds or diffusive fluxes, magnetic buoyancy may be able to sustain sufficient helicity fluxes to avoid quenching.

Equations

$$Ro({\tau }_{ed})=\tilde{R}o(\Omega )=\frac{1}{\Omega {\tau }_{ed}},$$

$$y(Ro)\equiv {\tau }_{cor}/{\tau }_{ed0},$$

$${\tau }_s=\frac{\Delta r}{r\Delta \Omega }=\frac{\Delta r}{r\Delta r{\partial }_r\Omega }=\frac{1}{q\Omega }=\frac{{\tau }_{ed0}}{q}$$

$$y=\left\{\begin{array}{rlr}& 1/q& Ro\to 0\\ & 1/Ro& Ro\to \infty \end{array}\right\}.$$

$${\tau }_{cor}^{-1}={\tau }_{ed}^{-1}+{\tau }_s^{-1},$$

$$y=\frac{1}{Ro+q}.$$

$$\frac{\rho v^2}{{\tau }_{cor}}=\frac{E}{{\tau }_{ed}{l}_{ed}^3}+\frac{\epsilon }{{\tau }_s}$$

$$\epsilon =\xi \rho (l_{ed0}/{\tau }_s{)}^2=\xi \rho l_{ed0}^2{q}^2{\tau }_{ed0}^{-2}=\xi q^2\rho v_0^2$$

$$\frac{\Gamma }{{\tau }_{ed}^{-1}}\simeq \frac{V}{l_{ed}^3}$$

$$\frac{E\Gamma }{V}=\frac{E}{{\tau }_{ed}{l}_{ed}^3}.$$

$$\frac{0.1\rho v_0^2/{\tau }_s}{\rho v_0^2/{\tau }_{cor}}=\frac{0.1\rho v_0^2/({\tau }_{ed0}/q)}{\rho v_0^2/({\tau }_{ed0}y)}=\xi q^{3}y=\frac{1}{20}$$

$$E/\rho \simeq v_0^5{\tau }_{ed0}^3$$

$$\frac{f^2}{y}=\frac{R{o}^4}{f^3}+\frac{q^3}{10},$$

$$f(Ro)\equiv v/v_0.$$

$$f=\left\{\begin{array}{rlr}& (R{o}^{4}y{)}^{1/5}& Ro\ge 0.22\\ & (q^{3}y/10{)}^{1/2}& Ro\le 0.22\end{array}\right\}.$$

$${\alpha }_0\sim {\xi }_{\alpha }{\tau }_{cor}^2{v}^2\Omega /h$$

$$y^2{f}^{3/2}\le \frac{h_0}{v_0{\tau }_{ed0}}=5$$

$$\frac{2h}{l_{cor}}=\frac{2h_0{f}^{-1/2}}{l_{ed0}fy}=10f^{-3/2}{y}^{-1}$$

$$\tilde{y}={\tau }_{cor}/{\tau }_{ed}.$$

$$\tilde{y}=\left\{\begin{array}{rlr}& \widetilde{Ro}/q& \widetilde{Ro}\to 0\\ & 1& \widetilde{Ro}\to \infty \end{array}\right\}.$$

$$\tilde{y}=\frac{\widetilde{Ro}}{\widetilde{Ro}+q}.$$

$$\epsilon =\xi q^2\rho v_0^2{\widetilde{Ro}}^{-2}$$

$$\frac{{\tilde{f}}^2}{\tilde{y}}=\frac{1}{{\tilde{f}}^3}+\frac{q^3}{10{\widetilde{Ro}}^3}$$

$$\tilde{f}=\left\{\begin{array}{rlr}& (\tilde{y}{)}^{1/5}& \widetilde{Ro}\ge 0.355\\ & (q^3\tilde{y}/10{\widetilde{Ro}}^3{)}^{1/2}& \widetilde{Ro}\le 0.355\end{array}\right\}$$

$${\tilde{y}}^2{\tilde{f}}^{3/2}/\widetilde{Ro}\le 5,$$

$$T_c={\left(\frac{\pi l_{ed}/4}{\Omega v}\right)}^{1/2}\simeq {\tau }_{ed}R{o}^{1/2}$$

$$\frac{{\tau }_{cor}}{{\tau }_{ed}}=\text{min}\{{\widetilde{Ro}}^{1/2},\tilde{y}(Ro)\}.$$

$$\frac{{\tau }_{cor}}{{\tau }_{ed0}}=\text{min}\{R{o}^{-1/2},y(Ro)\}.$$

$$\widetilde{Ro}=\frac{1}{{\tau }_{ed0}\Omega }=\frac{1}{{\tau }_{ed0}{\Omega }_0(r/r_0{)}^{-q}}={\left(\frac{r}{r_0}\right)}^q$$

$${\tau }_{ed}=\frac{r}{r_0}{\tau }_{ed0}\propto r$$

Full text

††pagerange: Some consequences of shear on galactic dynamos with helicity fluxes–LABEL:lastpage

1 Introduction

1.1 Background

In situ galactic dynamo theory has long been a leading paradigm to explain the ordered large scale magnetic fields of galaxies (Ruzmaikin et al., 1988). In this paradigm, a weak seed field, perhaps supplied primordially, is amplified via the action of turbulence and differential rotation in the galactic interstellar medium. How such dynamos work in detail, has been a longstanding research enterprise (Ruzmaikin et al., 1988; Brandenburg & Subramanian, 2005; Shukurov et al., 2006; Hanasz et al., 2009; Chamandy et al., 2014; Blackman, 2015; Kulsrud, 2015).

Standard (20th century) mean field $\alpha-\Omega$ galactic dynamos typically have at least three key ingredients (1) supernova driven turbulence, which in the presence of galactic rotation and stratification produces a kinetic helicity driven ”$\alpha$” effect that converts toroidal to poloidal field and (2) differential rotation that shears the poloidal field into the toroidal direction and (3) some kind of turbulent diffusion or loss term of the mean field in a thin disk that ensures the that the net toroidal flux in the disk reflects the observed field geometry (e.g. quadrupole).

A challenge for 20th century galactic dynamo theory has been the absence of a physical understanding of how the dynamo saturates. That basic theory is kinematic, considering only the growth of the large scale field without including the dynamics of the field on the driving flow. Intertwined with this deficiency has been the realization that standard mean field textbook $\alpha-\Omega$ dynamos also do not conserve magnetic helicity (Blackman & Field, 2000; Vishniac & Cho, 2001). (For reviews see Brandenburg & Subramanian (2005) and Blackman (2015)).

Principles of dynamically including magnetic helicity conservation in MHD turbulence from Pouquet et al. (1976) and modified lessons from steady-state mean field considerations of Gruzinov & Diamond (1994) and Bhattacharjee & Yuan (1995) were synthesized into time-dependent mean field dynamical toy models (Blackman & Field, 2002) using a simple closure (now referred to as ”minimal $\tau$”). In these models, the growth of a helical component of the large scale field is accompanied by growth of the oppositely signed small scale helical field which in turn, represents a backreaction that saturates the dynamo. For dynamos without shear this leads to a steady state, but for dynamos with shear, this can lead to catastrophic quenching alleviated only when helicity fluxes carry away the excess small scale field. Ultimately this requires a dynamo sustained by helicity fluxes (Blackman & Field (2000)). Depending on which terms in the electromotive force actually dominate, a complementary perspective is that the large scale dynamo is sustained directly via helicity fluxes even in the absence of any kinetic helicity (e.g. Vishniac & Cho (2001); Vishniac & Shapovalov (2014)). Helicity flux driven dynamos are conceptually related to the sustenance of large scale fields in the different context of laboratory magnetically dominated plasmas (Strauss, 1985; Bhattacharjee & Hameiri, 1986).

Incorporation of some these principles has led the numerical demonstration of the helpful role of magnetic helicity in numerical simulations of dynamos in stellar contexts (Brandenburg & Sandin, 2004; Chatterjee et al., 2011) as well as practical galactic dynamo models with helicity fluxes (Shukurov et al., 2006; Sur et al., 2007; Chamandy, 2016).

A second challenge of galactic and mean field dynamo theory is to incorporate the influences of rotation and shear on the turbulence, dynamo coefficients, and EMF. One approach is to expand the turbulent quantities into a base state that is independent of shear and rotation plus corrections that depend on them. The resultant mean turbulent EMF (whose curl enters the growth if the mean magnetic field) can then be expanded into a sum of all possible terms that are linear in the mean magnetic field and linear in the mean rotation or shear (Brandenburg & Subramanian, 2005; Rädler & Stepanov, 2006). The relevance and interpretation of each of these terms must be assessed independently for a given circumstance. However, this approach does not capture all of the effects of rotation and shear to all orders. Doing so formally is impractical, but physical approaches can provide insight and shortcuts.

1.2 Strategy and Outline

The influence of rotation can be partly gauged by the ratio of the nonlinear term in the Navier Stokes equation to the Coriolis term in the rotating frame. This dimensionless ratio, the Rossby number, is given by

[TABLE]

where $\Omega$ is the rotation speed and $\tau_{ed}$ is the eddy turnover time, presently defined in terms of the turbulence supplied specifically by supernovae. The latter is important to keep in mind as we will also utilize a separate correlation time $\tau_{cor}$ not necessarily equal to $\tau_{ed}$. The above equation introduces our convention of writing $Ro$ for the Rossby number for fixed $\Omega$ allowing $\tau_{ed}$ to vary and $\tilde{R}o$ for the Rossby number at fixed $\tau_{ed}$ allowing $\Omega$ to vary. How dynamos depend separately on $\tau_{ed}$, $\Omega$ and on differential rotation is not completely understood. Even for the basic $\alpha-\Omega$ type dynamo, the question of how the kinetic component of the helicity coefficient contribution $\alpha_{0}$ depends on rotation and shear warrants revisiting for strong shear.

There are a few precursors in this context. Ruediger (1978) calculated an effect of rotational quenching on $\alpha$. Ruzmaikin et al. (1988) considered the effect of the Coriolis force without shear and their prescription for the effect of rotation on $\alpha_{0}$ can be recast by replacing the correlation time of the turbulence $\tau_{c}=\tau_{ed}Ro^{-1/2}$ when $Ro\geq 1$, and $\tau_{c}=\tau_{ed}$ otherwise. In Chamandy et al. (2016), the same resulting piecewise-defined $\alpha$ was used. Blackman & Thomas (2015) and Blackman & Owen (2016) included an effect of shear on the correlation time by arguing that $\tau_{cor}$ equals $\tau_{ed}$ times a factor that depends on $Ro$ and shear.

In the present paper we explore and generalize a physical model for the influence of shear and rotation on both $\tau_{cor}$ and the turbulent energy density for galactic dynamos. We will see that when $Ro,{\tilde{R}o}>>1$, the supernova turbulence dominates both the turbulent energy density and its correlation time. In the regime $Ro,{\tilde{R}o}<<1$ shear can dominate both of these quantities. We build our model in three separate ways, first fixing $\Omega$ and changing $\tau_{ed}$, and then fixing $\tau_{ed}$ and changing $\Omega$. Then we consider a model in which they are mutually dependent on radius, via their connection to the star formation rate. The need for this arises because the dynamo depends separately on those two parameters not just in their dimensionless combination of the Rossby number. We explicitly derive $\tau_{ed}$ in terms of the SN rate and show how $\tau_{cor}$ changes as a function of rotation and shear. Both the effect of shear on the turbulent correlation time and as a supplemental source of turbulent energy have not been included in galactic dynamo models, although in the absence of SN, shear is expected to be a source of galactic turbulence (Sellwood & Balbus, 1999).

We also incorporate a magnetic buoyancy (MB) term (Parker, 1966) in the helicity flux term of the dynamo equations, generalizing the corresponding therm of Sur et al. (2007) which included only an advective wind flux term. The buoyant speed itself depends on the magnetic field, which increases the nonlinearity of the dynamo equations.

The paper is arranged as follows. In Sec. 2 we relate the turbulent velocity and correlation time to the Rossby number in both fixed-$\tau_{ed}$ and fixed-$\Omega$ cases, developing expressions for both the turbulent energy density and correlation time as a function of shear, rotation, and SN rate. For a given shear profile we consider three cases: (i) fixed $\Omega$, varying $\tau_{ed}$; (ii) fixed $\tau_{ed}$, varying $\Omega$; and (iii) mutually dependent variation of $\Omega$ and $\tau_{ed}$. We apply these relations to the dynamo equations in Sec. 3. The solutions are found numerically in Sec. 4, where we show both steady state solutions and time evolution of the magnetic fields. We identify where the results from our calculations that include the new ingredients differ from previous approaches. We also discuss the influence of magnetic buoyancy and the consequences of our calculations for observed pitch angle. We conclude in Sec. 5.

2 Effect of shear on correlation time and turbulent energy

The Rossby number is function of two variables $(\tau_{ed},\Omega)$. The value of $\tau_{ed}$ can vary for different supernova rates and $\Omega$ depends on the details of galaxy formation and the mass therein. In practice, these two quantities could be correlated because a fixed initial mass function for stars, and a baryon mass correlated with total mass would increase both the rate of SN and the rotation rate at a fixed radius. Below we consider separately cases where we allow these two quantities to be independent and then consider a case where they are mutually dependent. When they are independent, the dynamo then depends on these two variables independently, not just their ratio.

We first construct a physical model for the influence of shear on the turbulent energy and correlation time by fixing $\Omega$ and allowing $\tau_{ed}$ to change. We then construct the analogue where we keep $\tau_{ed}$ and allowing $\Omega$ to change. We show in the appendix that these two approaches can be unified. In the last subsection of this section we consider the case where the two quantites are mutually dependent.

In what follows, quantities with a subscript [math] (e.g., $\tau_{ed0}$, $v_{0}$, $l_{ed0}$ and so on) are evaluated at their fiducial values such that $Ro_{0}=\tilde{Ro}_{0}=(\tau_{ed0}\Omega_{0})^{-1}=1$.

2.1 Correlation time and turbulent energy for fixed $\Omega$, fixed shear, but different SN rates

2.1.1 Effect of the shear on the correlation time

We distinguish between the turbulent correlation time $\tau_{cor}$ and the naked eddy turnover time $\tau_{ed}$ determined by SN in the absence of shear, and $\tau_{ed0}$ as the fiducial value of the latter. We define the ratio of the former to latter as

[TABLE]

where $Ro=Ro(\tau_{ed})$ for fixed $\Omega$ in this section. The quantity $\tau_{cor}$ must satisfy the physically expected behaviors in the low and high $Ro$ limits, namely $\tau_{cor}\rightarrow\tau_{ed}$ as $Ro\rightarrow\infty$ and $\tau_{cor}\rightarrow\tau_{s}$ as $Ro\rightarrow 0$, where $\tau_{s}$ is defined as

[TABLE]

along with the rotation profile $\Omega\propto r^{-q}$. The physical meaning of $\tau_{s}$ is evident if we consider radially separated points on two concentric rings orbiting in the galaxy with radii $r-\Delta r/2$ and $r+\Delta r/2$ respectively. Their relative velocity will be $r\Delta\Omega=r\Delta r\partial_{r}\Omega$, and $\tau_{s}$ characterizes the time scale for these points to further separate to by $\Delta r$ in the azimuthal direction. In terms of $y$, the aforementioned asymptotic limits imply

[TABLE]

Deriving $y$ from first principles is a challenging endeavor but we can make good progress with a physically motivated approach. We posit that quadratic time correlations of turbulent quantities decay exponentially in time over a correlation time that has separate independent exponential factors from shear and SN turbulence. Then

[TABLE]

or equivalently,

[TABLE]

Eq. (6) satisfies the constraint (4). In the fast rotation limit $Ro\rightarrow 0$, we have $y=1/q$ so that Eq. (28) predicts a correlation time that asymptotically approaches a constant for $q>0$, as we will see below.

2.1.2 Effect of the shear on the turbulent energy

Next, we consider the effect of the shear on the turbulent energy. Technically the turbulent energy consists of both energy from supernovae and differential rotation since rotating MHD shear flows with $q>0$ are unstable (Velikhov, 1959; Balbus & Hawley, 1991). In terms of energy density input rate, this implies

[TABLE]

where $\rho$ is the average density of ISM, $v$ is the mean square root velocity of the turbulence, $E$ is the energy input to the ISM per supernova, $l_{ed}=v\tau_{ed}$ is the eddy scale, and $\varepsilon$ is the energy density input by shear and is taken to be a fraction of the fiducial shear energy density $\rho v_{s0}^{2}=\rho(l_{ed0}/\tau_{s})^{2}$. More specifically, we then have

[TABLE]

where we take $\xi=0.1$. To provide physical meaning for the second term in Eq. (7), we note that the energy density supplied by SN per unit time can be expressed as $E\Gamma V^{-1}$ where $V$ is the volume of the galaxy, and $\Gamma$ the rate at which SNe are produced in $V$. Crudely assuming SN occur isotropically, we have

[TABLE]

where $l_{ed}^{3}$ indicates the turbulent correlation scale from SN. Therefore

[TABLE]

We further assume that $E$ is a constant, and that the variation of $\rho^{1/3}$ with $Ro$ is small compared to $v$, so that $\rho$ can be taken as approximately constant as well. For the fiducial point values, the ratio between the second term on the RHS to the LHS of Eq. (7) is

[TABLE]

in using (2) and (6). For a flat rotation profile (as in typical spiral galaxies), $q\simeq 1$ so that this ratio is small and at the fiducial point values we can neglect the second term on the RHS of Eq. (7) to obtain

[TABLE]

which can then be used to simplify Eq. (7) to

[TABLE]

where

[TABLE]

Eq. (13) determines the nonlinear relation between the turbulent speed $v$ and $Ro$. However its solution has simple asymptotic behaviors. In the large $Ro\gg 1$ regime, $\tau_{ed}\rightarrow 0$ and SN energy input rate dominates over that of the shear, so we can drop the second term in Eq. (13) which leads to $f=(Ro^{4}y)^{1/5}$. In the $Ro\ll 1$ regime, the second term of Eq. (13) dominates so we drop the first term the right to obtain get $f=(q^{3}y/10)^{1/2}$. The two terms contribute equally at $Ro=0.22$ so we approximate $f$ as

[TABLE]

These relations capture the fact that as SN become scarce, the average turbulent speed of the ISM would decrease, but since shear provides a fixed baseline of turbulent energy, $v$ approaches to a constant.

Given Eq. (15) we are poised to check one more plausibility condition for $y$, namely that the magnitude of $\alpha_{0}\sim\tau_{cor}\langle\vec{v}\cdot\nabla\times\vec{v}\rangle$ cannot be larger than $v$, since the helical fraction cannot exceed unity. For quasi-isotropic turbulence (Durney & Robinson, 1982; Ruzmaikin et al., 1988)

[TABLE]

where $\xi_{\alpha}$ is a factor smaller than one and $h\propto v^{-1/2}$ is half of the scale height of the galaxy in an isothermal self-gravitating slab model (Spitzer, 1981). The required inequality is then $\tau_{cor}^{2}v^{2}\Omega/h\leq v$, or equivalently,

[TABLE]

upon using fiducial values $h_{0}=0.5\ \text{kpc}$, $\tau_{ed0}=10^{15}\ \text{s}$, and $v_{0}=10^{6}\ \text{cm/s}$, the validity of which can checked using Eqs. (6) and (15).

Note also that before turbulent energy is taken over by shear as $Ro$ decreasing, $l_{cor}=v\tau_{cor}$ will never exceed the scale height $2h$, since

[TABLE]

and it can be verified the quantity above is always greater than unity using (15).

2.2 Correlation time and turbulent energy for fixed SN rates but

different rotation rates

Complementing the previous subsection, here we instead fix the SN rate (and thus $\tau_{ed}$) but allow for different rotation rates. By direct analogy to (6), we define $\tilde{y}$ by

[TABLE]

Note that we have $\tau_{ed}=\tau_{ed0}$ here. The asymptotic limits are now

[TABLE]

By analogy to Eq. (4) we take

[TABLE]

For the turbulent energy, we now generalize the energy input from shear to allow varying angular velocity, assuming a fixed fraction is available. Thus Eq. (8) is replaced by (with $\xi=0.1$)

[TABLE]

which gives the correct value of $\varepsilon$ at the fiducial point where $\tilde{Ro}=Ro_{0}=1$. Now $\tilde{f}=v/v_{0}$ is given by

[TABLE]

of which the solution is approximately

[TABLE]

The plausibility analogue to Eq. (17) becomes

[TABLE]

which is also satisfied if we use Eqs. (21) and (24), and same fiducial values $h_{0}=0.5\ \text{kpc}$, $\tau_{ed0}=10^{15}\ \text{s}$, and $v_{0}=10^{6}\ \text{cm/s}$ as in the last subsection.

2.3 Generalizing the correlation time to

include the case of rotation without shear

For rigid rotation, $q\rightarrow 0$, and $y=Ro^{-1}$ yields $\tau_{cor}=\tau_{ed}$ as expected in the absence of shear. The effect of rigid rotation without shear on $\alpha_{0}$ has been previously considered due to the Coriolis acceleration (see p.163 in Ruzmaikin et al. (1988)). We can interpret this effect as a change to the correlation time as follows: Over a time $\Delta t$, the displacement from the Coriolis force can be estimated to be $d\sim\Omega v(\Delta t)^{2}$ and then we can set $\Delta t=T_{c}$ as the time inteval for the Coriolis force to rotate an eddy of radius $l_{ed}/2$ by $\pi/2$, or cause a displacement $d=\pi l_{ed}/4$. This $T_{c}$ is the time scale for two adjacent eddies to mutually shred from only this interaction and if this is the shortest of the eddy destruction mechanisms it would determine the correlation time. Using the above expressions for $d$ and $\Delta t$, we obtain

[TABLE]

Combining this with case of Sec. 2.2 (fixed $\tau_{ed}$), we can then write

[TABLE]

For the case of Sec. 2.1 (fixed $\Omega$), but with $q=0$, we can write $T_{c}=\tau_{ed0}Ro^{-1/2}$ and then

[TABLE]

Note that the Eqs. (27) and (28) incorporate the separate influences on the correlation time from pure rotation and shear. Fig. 1 shows the correlation time in our approach.

We may express $\tilde{Ro}$ as a function of the radial coordinate $r$ given the rotation profile, i.e.,

[TABLE]

where we have used $\tau_{ed0}\Omega_{0}=1$. Replacing $\tilde{Ro}$ by $r$ using the relation above and assuming all other variables are independent of $r$ provides us with one of the simplest way to write down a $r$-dependent model.

2.4 Case when correlation time and $\Omega$ both depend on $r$

In the cases considered above, we have assumed that the eddy time and the rotation periods are independent but in practice, models of star formation rates (SFR) in galaxies both depend on radius. We now suppose that $\tau_{ed}$ varies with the radial coordinate $r$ according the the prescription adopted by Prasad & Mangalam (2016). Specifically, we adopt the relation

[TABLE]

where $r_{0}=8\ \text{kpc}$ and it is determined from the following argument: if $\tau_{ed}^{-1}$ is proportional to the SN rate and the SN rate is proportional to the surface density of the SFR $\Sigma_{SFR}$ (Shukurov, 2004; Rodrigues et al., 2015), we have $\tau_{ed}^{-1}\propto\Sigma_{SFR}$. Further, we assume a Schmidt-Kennicutt-like power-law relation $\Sigma_{SFR}\propto\Sigma^{\xi_{g}}_{g}$ (Schmidt, 1959; Kennicutt, 1989; Heiderman et al., 2010), where $\Sigma_{g}$ is the gas surface density and typically $1\leq{\xi_{g}}\leq 1.4$. For simplicity, we take ${\xi_{g}}=1$ here. The mean galactic gas surface density $\Sigma_{g}\propto 1/r$ if the gas surface density hovers around a fixed fraction of order unity near the critical Toomre density for gravitational stability (Toomre, 1964; Cowie, 1981). Then combining these above relations we arrive at Eq. (30) above. If the helicity flux is driven by a galactic fountain, which in turn is driven by SN (Tenorio-Tagle & Bodenheimer, 1988; Shapiro & Field, 1976; Shukurov, 2004; Rodrigues et al., 2015) , we might consider that the outflow speed also satisfies

[TABLE]

In addition, for a flat rotation curve, $\Omega\propto 1/r$.

Now since both $\tau_{ed}$ and $\Omega$ vary with $r$, we need the unified relations derived in Appx. A, which results in

[TABLE]

and

[TABLE]

3 Dynamo Model for Low and High Rossby Numbers

The induction equation for the mean field is given by (for reviews Brandenburg & Subramanian (2005); Blackman (2015))

[TABLE]

where $\overline{\bf B}$ and $\overline{\bm{U}}$ are the (ensemble or spatial averaged) mean magnetic field and velocity field, respectively; $\overline{\bf J}=\bm{\nabla}\times\overline{\bf B}/\mu_{0}$ is the mean current (taking $\mu_{0}=1$); $\beta$ is the Ohmic resistive diffusion coefficient; \overline{\mbox{\boldmath{\cal E}}}{}=\alpha\overline{\bf B}-\beta_{t}\overline{\bf J} is the mean turbulent electromotive force where $\beta_{t}$ is the turbulent magnetic diffusivity, $\alpha\equiv\alpha_{0}+\alpha_{m}$ is the pseudoscalar helicity coefficient separated into kinetic and magnetic contributions, $\alpha_{0}$ and $\alpha_{m}$, respectively.

We adopt cylindrical coordinates and apply the ’no-z’ approximation (Subramanian & Mestel, 1993; Moss, 1995; Phillips, 2001; Sur et al., 2007) to reduce the PDEs to a simpler set of ODEs; the reduced dynamo equations read111Here we are working in the $\alpha\Omega$ dynamo approximation. For the more general $\alpha^{2}\Omega$ dynamo, an extra term $-2R_{\alpha}(1+\alpha_{m})B_{r}/\pi$ would appear on the right hand side in (36). This term is negligible compared to the term $R_{\omega}B_{r}$, since $|R_{\alpha}/R_{\omega}|\sim y^{2}Ro^{3\gamma}/15\ll 1$ using (39), and $|R_{\alpha}/R_{\omega}|\sim\tilde{y}^{2}/15\ll 1$ using (40), for all values of interest of $Ro$.

[TABLE]

where $B_{r}$ and $B_{\phi}$ are respectively the radial and azimuthal components of the total magnetic field. The $z$ component is assumed to be much less than these two and is neglected.

The second term on the RHS in (37) governs the effect of diffusive fluxes $\beta_{d}\nabla^{2}\alpha_{m}$ (Brandenburg et al., 2009a; Mitra et al., 2010; Hubbard & Brandenburg, 2010) where $\beta_{d}$ is the diffusion coefficient. For most of the discussion of the solutions in Sec. 4, we take $\beta_{d}=0$ (the case of Sur et al. (2007)) except for Sec. 4.5 where we adopt $\beta_{d}/\beta_{t}=1$ in a model using the radial coordinate $r$ as a free parameter and find that this diffusive helicity flux term raises the magnetic the saturated magnetic energy as it exceeds the wind flux term $R_{U}$ for the fiducial parameters chosen, over much of the disk. The last term in Eqn. (37) is the Vishniac-Cho flux (Vishniac & Cho, 2001) with dimensionless coefficient $\lambda_{V}$. We find that that, in accordance with Sur et al. (2007) that this flux has an influence only after the field already grows substantially, and has its strongest influence at low Rossby numbers. Even then, the buoyancy flux tempers the influence of the Vishniac-Cho flux. In the solutions presented in the sections below, we focus primarily the case of $\lambda_{V}=0$.

The magnetic fields are normalized by the equipartition field strength $B_{eq}=\sqrt{4\pi\rho}v$, so that $B_{r}=v_{A,r}/v$ and so on with $v_{A}$ the Alfvén speed. Note that $B_{eq}$ is a function of $v$ and thus varies with both the eddy turnover time and the galactic rotation speed. We normalize the time by the diffusion time scale $h^{2}/\beta_{t}$ which again depends on the Rossby number. The dimensionless parameters in the above dynamo equations are

[TABLE]

where $U$ is the buoyancy speed in $z$ direction containing both a convective flow part $U_{0}$ and a magnetic buoyancy part $U_{B}$; $\alpha$ is normalized by $\alpha_{0}$; and $l=v\tau_{cor}$ is the correlation length scale of the turbulence.

For the fixed-$\Omega$ case of Sec. 2.1, substituting (2) and (14) into those dimensionless parameters gives

[TABLE]

where $R_{U_{B}}$ is the magnetic buoyancy term which will be clarified later.

For the fixed-$\tau_{ed}$ case of Sec. 2.2, we use (19) and $\tilde{f}=v/v_{0}$ to obtain

[TABLE]

For the $r$-dependent model in Sec. 2.4 we use (59) along with $U\propto 1/r$ to get

[TABLE]

where $\tilde{r}=r/r_{0}$ with $r_{0}=8\ \text{kpc}$, and we use the following typical data for our Galaxy to calculate the fiducial values (same as in Sur et al. (2007), for the comparison later):

[TABLE]

which gives (with $q=1$)

[TABLE]

and the corresponding fiducial Rossby number $Ro_{0}\approx 1$.

The instantaneous dynamo number in the kinematic regime can be defined as the square of the ratio of the coefficients of the amplifying rate terms $\gamma_{g}$ and the decay rate terms $\gamma_{d}$:

[TABLE]

with

[TABLE]

and

[TABLE]

being respectively, the product of growth and decay terms in (35) and (36). We can define the dynamo growth time, divided by the diffusion time $\tau_{diff}=h^{2}/\beta_{t}$, as

[TABLE]

The bottom panel of Fig. 2 shows $\tau_{dyn}$ (thick blue line) in comparison with the age of the universe $\tau_{u}\approx 10^{3}\tau_{ed0}$, $\tau_{ed}$ and $\tau_{s}$ for our fixed-$\Omega$ case, while the dashed purple line ($\tilde{\tau}_{dyn}$) indicates the dynamo growth time for our fixed-$\tau_{ed}$ case. All times in the plot are normalized by $\tau_{diff}$. The vertical dot-dashed lines at $Ro=0.22$ and $Ro=0.355$ respectively, correspond to the transition values of Eqs. (15) and (24) respectively, and marking for each of these cases, the transition from shear dominated to supernova dominated turbulent velocities as $Ro$ increases. The top panel of Fig. 2 shows the 3-D space that unifies the the cases of Sec. 2.1 and 2.2 via Eq. 57.

Several interesting features are evident in the bottom panel of Fig. 2. First, in both two case, for either the fixed $\Omega$ or fixed $\tau_{ed}$, $\tau_{dyn}^{-1}\rightarrow 0$ when $Ro$ approaches $\sim 1.2$. As a consequence, for $Ro\gtrsim 1.2$, the initial growth of the magnetic field will be too slow to produce a significant large scale field. Second, $\tau_{dyn}$ becomes independent of $\tau_{ed}$ when $Ro\leq 0.1$ for fixed $\Omega$ (blue curve), because of the completely dominance of the shear as a supplier of turbulence. In contrast, for the case of fixed $\tau_{ed}$ (dashed purple line), the growth time blows up for $Ro\lesssim 0.02$ highlighting that field growth becomes insignificant at these values in this case. The top panel of Fig. 2 shows how these two different cases are mutually compatible in 3-D. The solutions further demonstrating these points will be discussed in the next section.

For the dynamo to have a significant influence on the large scale field, its growth time must be less than the age of the universe $\tau_{u}$. The associated condition $\tau_{dyn}\leq\tau_{u}$ leads to an upper bound on $Ro$ above which the dynamo solution cannot produce significant observable large scale fields . In addition, we impose a lower bound on $Ro$ for fixed $\tau_{ed}$ by the condition $\Omega_{\text{max}}r=c/10$ with $c$ the speed of light, simply so that we focus on the the cases where the rotation speed is non-relativistic. Combining these two constraints. we can express the physically meaningful range as

[TABLE]

for fixed $\Omega$, and

[TABLE]

for fixed $\tau_{ed}$.

4 Solutions

For the first 3 subsections below, we focus on the fixed-$\Omega$ case, before addressing a few important features of the fixed-$\tau_{ed}$ case in the penultimate subsection. In the last subsection we consider solutions for the case when $\Omega$ and $\tau_{ed}$ mutually depend on $r$.

4.1 Steady-State Without Magnetic Buoyancy

We first consider the case without magnetic buoyancy. By solving Eqs. (35)-(37) for a steady state ($\partial_{t}=0$) we obtain the darkest blue dotted line in Fig. 3. The $y$ axis, representing the magnetic field strength, is scaled with the equipartition field strength $B_{eq}^{2}=4\pi\rho v^{2}$ which depends on $Ro$. To the left of the the vertical dot-dashed line at $Ro=0.22$ the turbulence is mostly driven by shear and right to by SNe. The cusp irregularity at $Ro=0.22$ occurs because of our piecewise-defined (15), which in principle can be removed by rigourously solving $f$, but not essential for the level of detail explored here. We used (15), which is sufficient to capture the asymptotic behavior for large and small $Ro$. The darkest blue dotted-line solution includes the influence from differential rotation of both $\tau_{cor}$ and $v$ and can be compared with the top dotted line, obtained by taking for the full range of $Ro$

[TABLE]

which is the expression for $f$ that neglects the effect of shear in the turbulence and correlation time (though shear is still maintained for the $\Omega$ effect in the ${\overline{B}}_{\phi}$ equation).

In Fig. 4 we show how different components of the magnetic fields depends on the Rossby number. We define the pitch angle by

[TABLE]

where we have used (36) for the last equality. The magnitude of $p$ decreases with decreasing $Ro$ when the turbulent energy is mostly provided by SN (region to the right to the vertical line), in agreement with the numerical solution in Chamandy et al. (2016) (see their Fig. 2, where they used the Coriolis number $Co=1/Ro$). As expected, the pitch angle goes to a constant as $Ro\rightarrow 0$, since without SN, the turbulent energy and the correlation time depend only on the rotation profile. Then $Ro$ drops out of the equations and the dynamo saturates to a state purely driven by shear at fixed $q$. The smallness of the pitch angle is consistent with the basic observation that galactic magnetic fields are predominantly azimuthal Beck & Wielebinski (2013).

4.2 Role of Magnetic Buoyancy vs. Outflow and Diffusive Flux

We now investigate the inclusion of magnetic buoyancy. Although Foglizzo & Tagger (1994) suggest that differential rotation will stabilize the Parker mode, we neglect this effect in our rough calculations here. We use the buoyancy speed as calculated in Parker (1979). For a weak magnetic field of sub-equipartiation (with the turbulence) strength, $U_{B}\approx v_{A}^{2}/v=v(B_{r}^{2}+B_{\phi}^{2})$. For a magnetic field comparable to equipartition strength, $U_{B}\approx v_{A}=v\sqrt{B_{r}^{2}+B_{\phi}^{2}}$. The field-related buoyancy coefficient (assuming $|B_{\phi}|\gg|B_{r}|$) is then

[TABLE]

for fixed $\Omega$. (For the case of fixed $\tau_{ed}$ we would just replace $y$ by $\tilde{y}$ and $f$ by $\tilde{f}$ in the above expression.)

MB extracts small scale magnetic helicity but also large scale fields. As a consequence, there is a competition between the loss of large scale field and benefit to amplification from small scale magnetic helicity removal. The bottom dotted line of Fig. 3 shows the solution for the fixed $\Omega$ case. Here the presence of MB lowers the overall field strength compared to the case when $R_{U_{B}}=0$. For fixed $\Omega$, we also note the possibility of dynamo purely supported by only magnetic buoyancy, where $U_{0}=R_{U_{0}}=0$. This solution is represented by the lightest blue curve in Fig. 3.

The curves represented by green diamonds and red triangles of Fig. 4 show the different behaviors of the toroidal and poloidal magnetic fields. The growth of toroidal field ($B_{\phi}$, blue circles and green diamonds) is suppressed by MB, whereas the poloidal field ($B_{r}$, yellow squares and red triangles) is amplified by MB. This is understandable by noting the competing roles of MB mentioned above, and the fact that in Eqs. (35) and (36), MB is more significant for the toroidal field loss because $|B_{\phi}|\gg|B_{r}|$.

The importance of the diffusive helicity flux (second term on the right of Eq. (37)) can be assessed by its separate ratios to the wind term (first term on the right of Eq. (37) and the MB (third term on the right of Eq. (37)). For $\beta_{t}=\beta_{d}$ these are respectively

[TABLE]

and

[TABLE]

in the case of fixed $\Omega$. Since both ratios are smaller than 1 when $Ro<1$, keeping or neglecting the diffusive helicity flux term will not change the results significantly.

The pitch angle profile under the influence of MB is shown in Fig. 4. this curve explicitly reveals that MB more strongly suppresses azimuthal fields.

For this model, we can predict the tangent of the pitch angle as a function of $Ro$. The result is shown in Fig. 5 where we compare our numerical prediction with that of Chamandy et al. (2016) (who found $\tan p\sim\tau_{c}(v/h)^{2}/(q\Omega)$). The red part shows a power law $\ln(-\tan p)=1.13\ln(Ro)+constant$. The limited data in Van Eck et al. (2015) from their Fig. 8 suggests a slope of 0.4-0.5, if we assume that the surface SFR density $\propto$ surface SNR density $\propto 1/\tau_{ed}\propto Ro$. This is closer to the predicted value of Chamandy (2016) than ours, but more data and work are ultimately needed to pin down the tightness of these trends and predictions.

4.3 Time-dependent solutions

We now compare the time evolution of magnetic fields from the dynamo solutions for different values of $Ro$ in Fig. 6. The time is normalized by the constant $\tau_{r}=2\pi/\Omega$, and the magnetic fields are normalized by the ($Ro$-dependent) equipartition field strength.

The two lower curves show the transition from decaying solutions to those with an asymptotic sustenance of a steady-state as $Ro$ is dialed below $\sim 1.2$. As $Ro$ is deceased downward from 1.25, the dynamo growth time deceases. The growth time reaches a minimum (the dotted curve, $Ro=0.6$) and then increases, finally saturating (the solid curve), in agreement with Fig. 2. The dashed black curve indicates the fiducial point $Ro=1$.

4.4 Fixing $\tau_{ed}$ and changing $\Omega$

Fig. 7 shows the dynamo solutions using the relation (21) and the corresponding non-dimensional parameters in the dynamo equations for the case of fixed $\tau_{ed}$ and varying $\Omega$. The vertical dot-dashed line marks the transition value $\tilde{Ro}=0.355$ between shear-dominated and SN dominated turbulence. The maximum steady-state field strength $\sim 0.02B_{eq}^{2}$ occurs at intermediate $\tilde{Ro}\sim 0.2$, and decays with $\tilde{Ro}$ for both lower and higher $\tilde{Ro}$. This contrasts the saturated steady states of Figs. 3 and 4 for small $Ro$ where we fixed $\Omega$ and allowed $\tau_{ed}$ to vary.

4.5 Dynamo solutions as function of radius when SN rate depends on $\Omega$

Fig. 8 shows the result in using the model discussed in Sec. 2.4 and (41). The horizontal axis is normalized by $r_{0}=8\ \text{kpc}$. Here we define $e_{turb}=\rho v^{2}=\rho_{0}v_{0}^{2}f^{2}/r$ and $e_{B}=B^{2}$ as the turbulent energy density and magnetic energy density, respectively, and show them in blue curves. The ISM mass density is assumed to have the same dependence on $r$ as the galactic surface density, i.e., $\rho\propto 1/r$. Red curves represents the model with (48) being used, i.e., it neglects the effect of shear on both correlation time and turbulent energy density. Beyond the galactic central region $r/r_{0}<0.2$ where a more sophisticated dynamo model is needed, we obtain a nearly flat profile for both turbulent and magnetic energy density in agreement with Fig. 20 of Beck & Wielebinski (2013).

Curiously if we compare the two dashed lines, i.e., the turbulent energy densities with and without considering the shear, the latter is above the former yet the former one includes energy sources from both SN and shear. This is not surprising if we realize that even though cooperating shear into the model increases the turbulent energy input rate, the correlation time is decreased at the same time, leading to a net effect of lowering $e_{turb}$ (see Eq. (7)).

The black curve of Fig. 8 represents the magnetic energy density if we take the diffusive helicity flux term into consideration. Here the diffusion coefficient $\beta_{d}$ is assumed to be equal to the turbulent diffusivity $\beta_{t}$, which may be an overestimate because usually the ratio $\beta_{d}/\beta_{t}$ is taken to be $\ll 1$, e.g. in Brandenburg et al. (2009b) it is 0.05, and in Mitra et al. (2010) a value of $\sim 0.3$ is found (albeit at very low $R_{M}$ compared to what would be appopriate for galaxies). Using (32), (33) and (51), we find that the contribution from the wind term (characterized by $R_{U}$) is comparable to that from the diffusive term (characterized by $\pi/2$) when $r<0.3$, and $\pi/2\gg R_{U}$ when $r>0.3$, showing a dominance of this diffusive helicity flux in almost the whole disk. The inclusion of $\beta_{d}$ increases the saturated value of magnetic energy by nearly an order of magnitude given our fiducial parameter choices.

5 Conclusions

We have generalized a 2-D “no-z” galactic dynamo model with helicity fluxes to include two effects of differential rotation beyond its role in the $\Omega$-effect which have not been previously combined in galactic dynamo models. First, differential rotation provides an additional energy source for ISM turbulence, beyond that of SN. Second, differential rotation can shred turbulent eddies, reducing the correlation time of the turbulence (Blackman & Thomas, 2015).

We have incorporated these effects and relaxed the commonly assumed equality between the correlation time and the SN driven eddy turnover time. We show that the effect of shear on the correlation time can be important even when shear does not dominate the turbulent energy. For low SN rates and strong shear, both effects are important. We separately studied the influence of differential rotation on the mean field dynamo solutions as a function of the SN input rate and the rotation period when these quantities are taken to be independent and also when they are proportional to each other. The latter would be expected from correlations of the SN rate with the star formation rate and in turn, the galactic surface density and rotation rate (Prasad & Mangalam, 2016).

Our solutions show that the observable steady-state mean field dynamo field strengths at low Rossby numbers are significantly lower than those when the correlation time is independent of shear. The model also predicts the pitch angle of the mean field, a measure of radial to toroidal field magnitude, and a clean quantity to compare with observations (Chamandy & Taylor, 2015). Unlike previous work, we have also included magnetic buoyancy as a contributor to the helicity fluxes which becomes most important when $Ro,\tilde{Ro}<1$. We find that dynamos for which the helicity fluxes are entirely determined by buoyancy are possible even in the absence of advective or diffusive fluxes.

We also considered a model (Sec. 2.4) where both $\tau_{ed}$ and $\Omega$ are functions of $r$. All dimensionless parameters were then reinterpreted as functions of $r$ only, as in (41). When our model is used in this way to explore radial dependence of quantities within a galaxy, we derived that the magnetic energy density profile is relatively flat in radius, consistent with observations (Beck & Wielebinski, 2013) and it is a result that serves as a test/consistency check for the model. The shape of the curve is sensitive to the Schnmdt-Kennicut index of Sec. 2.4. If we switch it from 1 to 1.4, the radius at which the steady-state magnetic energy density drops to zero will move from $r\sim 0.2$ to $r\sim 0.7$.

Earlier prescriptions for galactic dynamos with $Ro<1$ included only the effect of $\Omega$ on the reduction of correlation time in $\alpha_{0}$ (Ruzmaikin et al., 1988), and without explicitly including the role of shear as a source of energy for the turbulence. We showed herein that shear causes a further reduction in the correlation time not captured by the previous treatments and an $\alpha\propto\Omega^{-3/4}$ for fast rotation in the fixed $\tau_{ed}$ case (Sec 2.2). This is a weaker reduction for fast rotators than rotational quenching in the absence of the shear effects, which predicts $\alpha\propto\Omega^{-1}$ Ruediger (1978).

Our calculations herein focused only on two specific influences of the role of shear and we do not purport to have captured all of the effects of shear on the turbulence and we have not included all terms in the EMF that depend on rotation. There are also other approaches to helicity flux driven mean field dynamos that bypass the $\alpha$ coefficient altogether. Our point in this paper however to focus on specific effects on shear that have been understudied. Future work should incorporate and assess the relevance of lessons learned here in the derivation of other dynamo coefficients not presently considered.

Acknowledgments

We thank L. Chamandy and E. Vishniac for related stimulating discussions. We acknowledge support from grants HST-AR-13916.002 and NSF-AST1515648. EB also acknowledges the Kavli Institute for Theoretical Physics (KITP) USCB and associated support from grant NSF PHY-1125915.

Appendix A Unified treatment of Sec. 2.1 and 2.2

We can formally combine the two cases Sec. 2.1 and 2.2 in a single formalism by defining

[TABLE]

so that $Ro=\tilde{\tau}_{r}/\tilde{\tau}_{ed}$. Then we can write

[TABLE]

The the energy rate balance equation is

[TABLE]

which, using $F=v/v_{0}$, can then be expressed as

[TABLE]

The solution is approximately

[TABLE]

and it is related to $f$ (Eq. (15)) and $\tilde{f}$ (Eq. (24))in Sec. 2 through

[TABLE]

Non-dimensional parameters (defined in Sec. (3) then exhibit the following scalings:

[TABLE]

Fix-$\Omega$ and fix-$\tau_{ed}$ cases correspond to, respectively, taking $\tilde{\tau}_{r}=1,\tilde{\tau}_{ed}=Ro^{-1}$ and taking $\tilde{\tau}_{ed}=1,\tilde{\tau}_{r}=Ro$ in the above relations.